RBF Kernel Matrix (Vectorised)

Kernel methods (SVMs, Gaussian processes, kernel ridge) compare every pair of points with a kernel. The most common is the RBF (Gaussian) kernel:

K(x, z) = \exp\left(-\gamma \|x - z\|^2\right)

Implement rbf_kernel(X, Z, gamma=1.0) returning the (len(X), len(Z)) matrix without Python loops. Use the identity $\|x - z\|^2 = \|x\|^2 + \|z\|^2 - 2\,x \cdot z$ , and clip tiny negative values (from floating-point error) to 0.

Then implement kernel_ridge_predict(X_train, y_train, X_test, gamma=1.0, lam=1e-3): solve $(K + \lambda I)\alpha = y$ on the training data and predict $K_{test,train}\,\alpha$ .